Heat Kernel of Networks with Long-Range Interactions

Figure 12

Zeta function plotted against the exponent for the graph in Figure 7. Subfigure (a) displays the zeta function for the Laplace transform of the graph Laplacian with different values of (2, 2.5, 3, and 4). Subfigure (b) illustrates the zeta function for the Mellin transform of the graph Laplacian with various values of (2, 2.5, 3, and 4). By increasing and , we can see that the zeta functions tend to the zeta function of the standard Laplacian (blue curve). This occurs much faster when the Laplace transform is used (a) compared to the Mellin transform (b).

(a)

(b)